The Ellis Semigroup of a Generalised Morse System

Petra Staynova

Published 2017-11-28Version 1

In 1997, Haddad and Johnson prove that the Ellis semigroup of any generalised Morse sequence has two minimal ideals. In that paper, they first compute the idempotents in the $\mathbb{N}$-shift case and use the fact that the closure of the set of idempotents is precisely the set of IP cluster points, so in this case every IP set is an idempotent. Then, they use a proposition (given without proof) to extend this computation of the IP cluster points of the $\mathbb{N}$-shifts to a computation of the IP cluster points of the $\mathbb{Z}$-shifts. In this note, we provide a counterexample to this proposition, and prove the main theorem of Haddad and Johnson without using IP cluster points. We will augment and complement their ideas, using results from the well-known book of Ellis and Ellis, 'Automorphisms and Equivalence Relations in Topological Dynamics'. We will further explicitly describe the two ideals in terms of their idempotents.